Circulation and the Fast-Growing Hierarchy

[JmsNxn]

I don't see how that would relate to Fast Growing Hierarchies. Could you clarify your question, Catullus?

 F(x) > 2[+1]x. for alpha greater than 1. So f(x) would be greater than 2[+1]x. the super function of 2[]x or 2 omegated to the x. But on the Googology Wiki page for circulation (https://googology.fandom.com/wiki/Circulation) "Using only nonnegative integers, there are only four cases where circulation is defined:". Should not its super function at base two grow slower than f(x)? F(x) is defined using for lots of cases using only positive integers.

OHHHHHHH!!!!

This is a very fascinating question. I will gladly answer this. It took me awhile to understand what you meant. So let's write \(\uparrow f\), to represent "take the super function" of \(f\). The answer to your question is actually pretty dumb, as it comes from analysis. It's only interesting in iteration theory.

Let's take \(f\) and it's superfunction \(F\), such that \(F(s+1) = f(F(s))\). Now, let's assume \(f\) is constant. \(f=C\). Well then, \(F=C\).

\[
\uparrow \text{Constant} = \text{Constant}\\
\]

So what happens in this omegation instance, is that we either hit \(4\) when \(x=2\), we hit \(1\) when \(x=1\), we hit \(1\) when \(y=0\), or we hit \(\infty\). The value \(\infty\) in this case can be thought of as a constant. And:

\[
\uparrow \infty = \infty\\
\]

So all of these "omegations" are hitting a fixed point value. I think what you are trying to look at is a little bit different. You want to look at:

\[
2 \uparrow^{2\uparrow n}n\\
\]

And you are asking of super functions in this manner. That's the only way I can think of which introduces fast growing hierarchies...

I hope I'm in the ball park. It'd help if you worded out more what your asking... I half get what you are asking. Can you just write more, and explain further?

Regards, James